Device for driving and for the accurate and stable positioning of rotary element

ABSTRACT

The device according to the present invention comprises means for driving and positioning a rotary element whereby the latter is driven by at least one driving element in accordance with a cyclic program which can be performed in one or more revolutions of the driving element or elements and/or of the driven element, the device being adapted to operate in both directions of rotation, the program being composed, in succession and inalternation, of a homokinetic drive and of a nonhomokinetic drive, and the rotation of the driven element, measured by the rotation of the driving element, being any predetermined function of the program.

United States Patent lnventor Roger Marcel Chaveneaud Rouquier-Leavallois Perret, France 810,607 Mar. 26, 1969 July 6, 197 1 Societe Lamy DEtudes Et De Recherches Soler Courbevoie Rants de Seine, France Apr." 2, 1968 France Appl. No. Filed Patented Assignee Priority DEVICE FOR DRIVING AND FOR THE ACCURATE AND STABLE POSITIONlNG 0F ROTARY ELEMENT to cum, 13 Drawing Figs.

u.s.c| 74/820, 4 74/8 1, 74/435, 74/436, 74/437 1m. c1 B23q 17/18 FieldotSearch 74/820,

References Cited UNITED STATES PATENTS 9/1925 Drissner et a1 1,928,925 10/1933 Brophy 74/820 2,797,589 7/1957 Chavenaud..... 74/436 2,922,505 1/1960 De Buigne 74/820 X 2,986,949 6/1961 Lancaster et all 74/820 X 3,170,333 2/1965 Umbricht 74/84 3,408,877 11/1968 Olson 74/84 Primary Examiner-Milton Kaufman Assistant Examiner-F. D. Shoemaker Attorney-Finnegan, Henderson & Faralbow ABSTRACT: The device according to the present invention comprises means for driving and positioning a rotary element whereby the latter is driven by at least one driving element in accordance with a cyclic program which can be performed in one or more revolutions of the driving element or elements and/or of the driven element, the device being adapted to operate in both directions of rotation, the program being composed, in succession and inalternation, of a homokinetic drive and of a nonhomokinetic drive, and the rotation of the driven element, measured by the rotation of the driving element, being any predetermined function of the program.

PATENTED JUL 6 I97:

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DEVICE FUR DRIVING AND FOR THE ACCURATE AND STABLE PQSIITIONIING F ROTARY ELEMENT The present invention relates to devices for driving and for the accurate and stable positioning of rotating elements ac cording to a cyclic program.

Many mechanisms are already known which are designed to drive and position rotating elements, including in particular the so-called Maltese cross or Geneva stop mechanisms, as well as devices comprising a Maltese cross and juxtaposed toothed segments.

There are also known cam and roller devices and controlling devices for intermittent movements which have been further improved, such as that described in French Pat. No. 1,046,301 and its Pat. of Addition No. 63,550.

Each of these mechanisms offers various possibilities, but they are limited to a modest number of combinations.

It is an object of the present invention to provide a device which can be utilized in a wide variety of mechanisms, the drive of which in accordance with any cyclic program will always be extremely precise and free from slip, and in which the part to be positioned can be stopped with precision at all the points desired, these points being as close together as desired.

it is also an object of the present invention to provide a device which will enable the mechanism in which it is used to be operated at high speeds, without impacts, without slip, and in accordance with varied programs, the element to be driven being always driven positively, whereby considerable torques can be transmitted.

The device according to the present invention comprises means for driving and positioning a rotary element whereby the latter is driven by at least one driving element in accordance with a cyclic program which can be performed in one or more revolutions of the driving element or elements and/or of the driven element, the device being adapted to operate in both directions of rotation, the program being composed, in succession and in alternation, of a homokinetic drive and of a nonhomokinetic drive, and the rotation of the driven element, measured by the rotation of the driving element, being any predetermined function of the program.

The device according to the present invention is characterized in that it includes: at least one driven element comprising, on the one hand, at least one first freely rotatable bearing means, determining the drive and positioning of the said driven element in one direction of rotation, and, on the other hand, at least one second freely rotatable bearing means, determining the drive and positioning of the driven element in the reverse direction of rotation, and also gear teeth to effect a homokinetic drive of the driven element, the first and the second bearing means being situated in positions in which they and the said gear teeth effect together each termination and each commencement of homokinetic drive whereby the driven element is accurately driven and positioned without slip, by a positive drive without play or impacts; the first bean ing means remaining, throughout the positioning and the nonhomokinetic drive by the two bearing means, located on one side of a line joining the centers of the driven element(s) and driving element(s) and the second bearing means being located on the other side of the said centerline; and at least one driving element comprising, on the one hand, at least one continuous rolling surface cooperating with a first bearing means of the driven element and effecting in the first-mentioned direction of rotation and in a continuous manner'the drive and the positioning of the driven element, the said rolling surface being composed successively ofa homokineticdrive curved part, a curved part corresponding to a predetermined function of the program, and a curved part in the form of a tooth cavity corresponding to the taking up of the homokinetic drive, and on the other hand, at least one continuous rolling surface cooperating with a second bearing means of the driven element and effecting in the reverse direction of rotation and in a continuous manner the drive and the positioning of the driven element, this latter rolling surface being composed successively of a curve part affording a homokinetic-drive tooth cavity, a curve part corresponding to a predetermined function of the program, and a curve part corresponding to the taking up of the homokinetic drive, and gear teeth cooperating with the gear teeth of the driven element and situated on the driving element or elements in positions in which they and the two bearing means cooperating with the respective homokinetic-drive curved parts of the two conjugate rolling surfaces effect together each termination and each commencement of homokinetic drive. i

in this device, at least one first bearing means carried by the driven element is adapted to cooperate successively during one and the same program with at least two distinct continuous rolling surfaces, situated on at least two different sectors of the same driving element, limiting, in the direction of rotation, the drive and positioning of the driven element, and performing at least two nonhomolcineticfunctions of the program; at least one second bearing means carried by the same driven element being adapted to cooperate successively with at least two continuous rolling surfaces, each being conjugate with the corresponding (forward) continuous rolling surface and effecting, in the reverse direction of rotation, the drive and positioning of the driven element.

When the device comprises at least one driven element mounted on an axle and at least two driving elements mounted on two other axles, the driven element may carry at least two bearing means adapted to cooperate simultaneously with two driving elements mounted on different axles, each driving element comprising at least one continuous rolling surface, each of the said bearing means cooperating successively, as the program proceeds, with all the continuous rolling surfaces carried by the various driving elements, and performing the successive functions of a first bearing means determining in the (forward) direction of rotation the drive and positioning of the driven element, and then of a second bearing means determining in the reverse direction of rotation the drive and positioning of the said driven element.

When the device is intended to effect the stopping of the driven element at positions as near as possible to specific points, at least one first bearing means of the driven element may, to determine in the (forward) direction of rotation the positioning of the said driven element, be adapted to cooperate with at least two continuous rolling surfaces carried by the driving element or elements, the said rolling surfaces each comprising a part ofa cylindrical curve having a different radius from but the same center as the revolution axis of the driving element; and at least one second bearing means of the driven element may, to determine in the reverse direction of rotation the positioning of the said driven element, be adapted to cooperate with at least two continuous rolling surfaces carried by the driving element or elements, the said rolling surfaces each comprising a part of a cylindrical curve having a different radius from but the same center as the revolution axis of the driving element, each of the latter continuous rolling surfaces being conjugate with the corresponding first rolling surface.

The driven element may carry at least two pairs of first and second bearing means to cooperate each with at least one of the two pairs of conjugate rolling surfaces carried by the driving element or elements, each pair of bearing means being situated on a different segment of the driven element, the bearing means indexed in the same segment being in different planes (these being planes perpendicular to the axis of rotation of the driven element), and the driving element or elements affording at one and the same time both of the pairs of conjugate surfaces which are to cooperate with one of the pairs of bearing means of the driven element; whether a drive or a positioning function is selected will then be determined by the pair of bearing means presented by the driven element.

The axles of the driven element and of the driving element are not necessarily parallel. Thus, they may run towards each other, the first and the second bearing means carried by the driven element being in the shape of a truncated cone, the corresponding rolling surfaces of the driving element being conjugated to cooperate with the said bearing means in the shape of a truncated cone, and the homokinetic-drive teeth being con stituted by tapered bearing teeth.

The device may include one or more driving elements each cooperating with several driven elements.

The bearing means of the driven element may be supported by cheeks formed by toothed planes (or platelike components) of the said driven element.

Each first bearing means of the driven element may be mounted on the first tooth of each homokinetic-drive toothed segment and each second bearing means may be mounted on the last tooth of one of the said segments, the diameter of the bearing means and its position in relation to the nominal diameter of the homokinetic-drive teeth being such that the circumference of the said bearing means is tangential to an imaginary concave curve representing the side of a gear tooth.

To obtain nonhornokinetic functions, preceded and followed by a homokinetic drive with a different ratio, the diameter of the two bearing means of the driven element and their position may be such that the circumference of the said bearing means-will be tangential to an imaginary concave curve representing the side of a gear tooth the nominal diameter of which is that of the smaller of the two toothed segments.

The driven element may be made of resilient material, and may for example be a plastic moulding, the bearing means of the driven element being mounted on gear teeth having a resilient opening acting as an axle retainer, the said opening being directed in such a way that throughout the cooperation of the bearing means with a rolling surface, the forces acting upon the bearing means tend to impel it towards a rounded bottom of the said opening.

The invention will be described in greater detail with reference to the accompanying drawings in which:

FIG. I shows diagrammatically one embodiment of the present invention, seen in elevation,

FIG. 2 is a view similar to FIG. I, the driving and driven elements of the device being located differently,

FIG. 3 is a plan view of the device of FIGS. I and 2,

FIG. 4 is a diagram showing the driving program of the device of FIGS. I to 3,

FIG. 5 represents diagrammatically in elevation another embodiment of the present invention,

FIG. 6 is a plan view of the device of FIG. 5,

FIG. 7 shows diagrammatically in elevation another embodiment of the present invention,

FIG. 8 is a view similar to FIG. 7, the driving and driven elements of the device being differently located,

FIG. 9 is a plan view of the device of FIGS. 7 and 8,

FIG. 10 is a diagram representing the driving program of the device shown in FIGS. 7 and 8,

FIG. II shows an example of the mounting of the bearings on the driven element,

FIG. 12 is a plan view showing the arrangement of the bearings in parallel planes, and

FIG. 13 represents a form of mounting of a bearing on a tooth.

The device represented in FIGS. I and 2 comprises a driving element I with integral portions thereof disposed in three different parallel planes mounted on an axle 2 and comprises two different segments. The driving element is two pairs of conjugate rolling surfaces one of which includes 3 and 3' situated in one and the same plane and the other of which includes rolling surfaces 4 and 4 situated in another plane, as shown in FIG. 3. In addition, the driving element I has in its third plane two toothed homokinetic gear sectors 5 and 5.

The driving element I cooperates with a driven elemento, also having integral portions thereof disposed in three different parallel planes. The driven element 6 is mounted on a shaft 7 and supports, in a first of said planes, three bearings 3, 8 and 8" spaced at I and each aligned to cooperate, during its driving and positioning program, with the two rolling surfaces 3 and 3' of the driving element I. Each of the bearings 8, 8' and 8", during its cooperation with one of the said rolling surfaces 3, 3 effects the rotational driving and the positioning of the driven element 6 which carries in its second plane three other bearings 9, 9' and 9" also spaced at and aligned for cooperating with the surfaces 4 and 4 of the driving element 1, to effect the driving and establish the positioning of the driven element 6 in the reverse direction of rotatron.

The bearings 8, 8 and d are freely rotatably mounted (FIGS. I and 3) on respective axles 811, 8'1: and 8"(1, themselves supported by two cheeks I0 and II. The bearings 9, 9' and 9 are mounted on the respective axles 9a, 9'a and 9"a, supported by the cheeks II and I2, the cheeks III, II and 12 being integral with and carried by the driven element 6.

Each bearing 3, 8' or 8 remains, during its cooperation, located at the rear of the line of centers joining the axles 2 and 7 of the elements, each bearing 9, 9' or 9" being, during the same cooperation, situated forward of said line. Finally, the driven element 6 has in its third plane three toothed sectors I3, 13' and 13 for homokinetic driving, each aligned to cooperate in turn with one of the two toothed segments 5 and 5 of the driving element I.

In this example, during the program that is performed in two turns of the' driven element 6, and three revolutions of the driving element 1, each of the three bearings 8, 8' and 8" cooperates once with the rolling surface 3 which presents in turn to the said bearing: a part Id for homokinetic driving, a part 15 effecting deceleration, a concentric part I6 defining the stop of the driven element 6 at positions 0, 8 and I6, a part 17 efiecting acceleration and finally a part having the shape of a tooth cavity 18, accompanying the taking up of the homokinetic drive. Each of the three bearings 53, 8' and 8" cooperates also once during the same program with the rolling surface 3', similarly composed of a part I4 for homokinetic drive, a part (unnumbered) controlling the deceleration, a part 16 concentric with the axle 2 of the driving element I, of smaller diameter than that of the similar concentric part 16 of the surface 3, defining the stopping of the driven element 6 at positions I, 9 and I7, a part (unnumbered) effecting the acceleration, and finally a part having the shape of a tooth cavity 13', accompanying the taking up of the homokinetic drive.

Each of the three bearings h, 9, 9 cooperates once during the program with the rolling surface 4,, conjugate with the surface 3, the said surface 4 being composed successively of: a part having the shape of a tooth cavity I9 effecting the homokinetic drive, a part (unnumbered) controlling the deceleration, a part 20 concentric with axle 2, a part (unnumbered) effecting the acceleration, and a homokinetic driving part 2E. Each of the three bearings 9, 9', cooperates also once with the rolling surface 4 conjugate with the surface 3', the said surface 4' being similarly constituted by a homokinetic driving part 119', a part (unnumbered) controlling the deceleration, a part 2t) concentric with axle 2 and of greater radius than the similar part (unnumbered) of surface 41, a part (unnumbered) effecting the acceleration and a homokinetic drive part M.

The gearing teeth are arranged on the driving element I and the driven element 6, so that the beginning and the end or" each homokinetic drive period shall be simultaneously effected by the gearing teeth I3 and 5 and by the two bearings 8 and 9 cooperating with the homokinetic drive parts of the cor responding rolling surfaces 3 and 4. From this example it will be seen that two neighboring positions I and 2, 8 and 9 and I6 and I7 are determined by the two same bearings 8, 9, 8' and 8" and 9".

FIG. I represents the driving element I passing through position 23; the driven element 6 having completed its deceleration and having stopped at position 0.

FIG. 2 represents the driving element 1, in position 14; the driven element 6 having stopped at position I.

FIG. 4 shows in Cartesian coordinates an illustrative program in which the movement of a driven element as is measured by the rotation of a driving element 1 whose positions are shown as abscissae, and the positions of the driven element 6 are shown as ordinates.

FIG. 5 shows another example of embodiment of the device according to the invention: the driven element 6, mounted on a shaft 7 is controlled by two driving elements 22 and 23 mounted on the shafts 24 and 25. The driving elements 22 and 23 have respective teeth 26, 27 which are in mesh with teeth 28 of a driving pinion 29 mounted on a shaft 30.

In FIG. 6 four bearings 31, 32, 33 and 34 are mounted in a common plane between two cheeks 35 and 36 fixed onthe shaft 7 of the driven element 6.

The driven element 6 also carries four toothed segments 37, 38, 39, 40 for homokinetic driving, each cooperating alternately with the teeth 26 and 27 of the two driving elements 22 and 23. Each bearing 31, 32, 33, 34 cooperates alternately first with a continuous rolling surface 41 of the driving element 23 and then with a continuous rolling surface 42 of the driving element 22, to function first in the manner of the hearing 8, and then that of the bearing 9 of the example shown in FIG. 1. The bearing which cooperates with the driving element 23 limits the driving and the movement of the driven element 6 in the direction of rotation, whereas the bearing which cooperates with the driving element 22 determines the driving and the positioning of the driven element 6, in a reverse direction of rotation. In this example, even though each driving element 22 or 23 is disclosed as having only a continuous rolling surface 42 or 41 it is considered obvious that any other combination of the surfaces and of the bearings is possible without departing from the scope of the invention.

FIG. 7 represents another embodiment of the device according to the invention, comprising a driven element 43 mounted on a shaft 44, and a driving element 45 mounted on a shaft 46.

As is best shown in FIG. 9, the driven element 43 bears five cheeks 47, 48, 49, 50 and 51 between which are mounted four bearings 52, 53, 54, 55 situated in four different parallel planes aligned to cooperate with four rolling surfaces 56, 57, 58 and 59, respectively, carried by the driving element 45. Thus, it will be seen that said surfaces constitute two pairs of conjugate surfaces, each of those pairs being able only to cooperate with one of the two pairs of bearings.

In addition to the bearings 52 to 55, the driven element 43 carries, in a fifth plane, a toothed segment 60 and, in a sixth plane, a toothed segment 61., which segments can cooperate respectively with toothed segments 62 and 63 carried by the driving element 45.

The axles (unnumbered) of the four bearings which are car ried by the cheeks 47 to 51 are advantageously located on the nominal circumference of the smaller of the two toothed segments 60 and 61, in this example segment 60, so that the circumference of each bearing is tangential to an imaginary curve representing a side of a homokinetic gear tooth of the small toothed segment 60.

The driving element 45 is represented in FIGS. 7 and 8 in position 9, whereas, in FIG. 7, the driven element 43 is represented in position 12 and in FIG. 8, in position 27.

The diagram of the sequence of the relative positions of the two elements is shown in FIG. 10, where the ordinates show the, positions of the driven element 43 and the abscissae show the; positions of, the driving element 45. This diagram represents the programof the driving and positioning of the drivenelement 43.

Assaid FIG. shows, the cyclic driving and positioning program which is effected in one revolution of the driven element 43and in two revolutions of the driving element 45 is composed successively:

of a first nonhomokinetic function corresponding to the distance covered by the driven element 43, from its position 10 to its position 13, and comprising a deceleration from 10 to 12, a stop stabilized at position 12 and an acceleration from 12 to 13, by the cooperation of two respectively 56 and 57;

lit

of a drive from position 13 to position 27 in the ratio 32 24 by engagement of the homokinetic gear teeth 61 of the driven element 43 with the teeth 63 of the driving element 45;

of a second nonhomokinetic function corresponding to the distance covered by the driven element 43 from its position 27 to its position 31 and comprising a deceleration from 27 to 28, a nonhomokinetic drive by rolling, from 28 to 29, and an acceleration from position 29 to 31, through the cooperation of the two bearings 54 and 55 respectively with the two conjugate surfaces 58 and 59;

of a drive from position 31 to position 10 passing through position 0, in the ratio 16 24 by the engagement of the homokinetic gear teeth 60 of the driven element 43 with teeth 62 of the driving element 45.

FIGS. 11 and 12 represent a particularly simple and advantageous mounting of the bearings 64, 64', 65 and 65' on a driven element 66 controlled by a driving element 67. In this method of mounting, the bearings are located in three planes between cheeks 68, 69 and 70, integral with element 66 and each comprising two segments with homokinetic toothing aligned to engage with three toothed homokinetic segments 71, 72 and 73 of the driving element.

The diametrically opposed bearings 65 and 65 are situated between the cheeks 68 and 69 of the driven element 66, and the bearings 64 and 64' between cheeks 69 and 70. Bearings 64 and 64 are aligned to cooperate alternately with a rolling surface 74 formed on element 67 between checks 72 and 73. Similarly, rollers 65 and 65' are aligned to cooperate with a rolling surface 75 formed on element 67 between cheeks 71 and 72.

Bearings 65 and 65' are borne by the first tooth of each of the homokinetic gearing segments of cheeks 68 and 69 of the driven element 66 and bearings 64 and 64 are borne by the last tooth of each of the homokinetic gearing segments of cheeks 69 and 70. The first and last teeth of each of the segments are advantageously large teeth cooperating with checks 71, 72 and 73 ofthe driving element 67.

FIG. 13 shows another example of mounting any of the bearings on parts constructed of resilient material, for example, in moulded plastic, in which a tooth: 76 comprises a slit 77 orientated in such a way that the component of the forces during the cooperation of a bearing 78 with a rolling surface (not shownlforces the bearing toward and against a rounded bottom of said slit with the bearing 78 and its axle 79 being a single part.

When the driving and driven elements have to transmit appreciable torques, each hearing may be mounted on a ball or roller bearing, or be formed directly of such a bearing supported by an axle fixed in the parts of the driven element forming cheeks.

Modifications of shape and detail may be made to the various alternatives of embodiment of the device mentioned merely for the sake of nonlimitative example, without as a consequence of this departing from the spirit of the invention.

Iclaim:

l. A device for driving and positioning rotary elements comprising cooperative rotatable driving means and driven means, said driven means including at least a pair of cam followers and a tooth segment, said driving means including a tooth segment aligned for meshing engagement with said driven means tooth segment during at least a portion of the rotation of said driving and driven means, said driving means further including a pair of cams aligned for contacting respective ones of said cam followers, 3 first of said cams having a lobe defining a pair of cam surface means for respective driving engagement with one of said cam followers during rotation of said driving means in a first direction and for positioning engagement with said same cam follower during rotation of said driving means in a second and opposite direction, and a second of said cams having a lobe defining a pair of cam surface means for respective driving engagement with the other of said cam followers during the rotation of said driving means in said second direction and for positioning engagement with said other cam follower during rotation of said driving means in said first direction.

2. The device as defined in claim 1 wherein one of said cams and cam followers are in a first plane and another of said cams and cam followers are in a second plane.

3. The device as defined in claim 1 wherein one of said cams and cam followers are in a first plane and another of said cams and cam followers are in a second plane, and said first and second planes are in parallel spaced relationship.

4. The device as defined in claim 1 wherein said cams and cam followers are in a common plane.

5. The device as defined in claim 1 wherein said first cam additionally-has a second lobe contoured to the configuration of said second cam lobe, and said second cam additionally has a second lobe contoured to the configuration of said first cam lobe.

6. The device as defined in claim 1 wherein said tooth segments define a predetermined input-output gear ratio between said driving and driven means, and said driving and driven means include other aligned meshable tooth segments defining an input-output gear ratio different than said predetermined ratio.

7. The device as defined in claim 1 wherein'said driving and driven means each has an axis of rotation, and one of said pair of cam followers is positioned on a side of a line between said axes opposite to that of the other of said pair of cam followers.

8. The device as defined in claim 1 wherein means are provided for mounting said pair of cam followers upon teeth of said driven means tooth segment.

9. The device as defined in claim 2 wherein said tooth segments are in a third plane spaced from and parallel to said first and second planes.

10. The device as defined in claim 5 wherein one of said cams and cam followers are in a first plane and another of said cams and cam followers are in a second plane in spaced parallel relationship to said first plane. 

1. A device for driving and positioning rotary elements comprising cooperative rotatable driving means and driven means, said driven means including at least a pair of cam followers and a tooth segment, said driving means including a tooth segment aligned for meshing engagement with said driven means tooth segment during at least a portion of the rotation of said driving and driven means, said driving means further including a pair of cams aligned for contacting respective ones of said cam followers, a first of said cams having a lobe defining a pair of cam surface means for respective driving engagement with one of said cam followers during rotation of said driving means in a first direction and for positioning engagement with said same cam follower during rotation of said driving means in a second and opposite direction, and a second of said cams having a lobe defining a pair of cam surface means for respective driving engagement with the other of said cam followers during the rotation of said driving means in said second direction and for positioning engagement with said other cam follower during rotation of said driving means in said first direction.
 2. The device as defined in claim 1 wherein one of said cams and cam followers are in a first plane and another of said cams and cam followers are in a second plane.
 3. The device as defined in claim 1 wherein one of said cams and cam followers are in a first plane and another of said cams and cam followers are in a second plane, and said first and second planes are in parallel spaced relationship.
 4. The device as defined in claim 1 wherein said cams and cam followers are in a common plane.
 5. The device as defined in claim 1 wherein said first cam additionally has a second lobe contoured to the configuration of said second cam lobe, and said second cam additionally has a second lobe contoured to the configuration of said first cam lobe.
 6. The device as defined in claim 1 wherein said tooth segments define a predetermined input-output gear ratio between said driving and driven means, and said driving and driven means include other aligned meshable tooth segments defining an input-output gear ratio different than said predetermined ratio.
 7. The device as defined in claim 1 wherein said driving and driven means each has an axis of rotation, and one of said pair of cam followers is positioned on a side of a line between said axes opposite to that of the other of said pair of cam followErs.
 8. The device as defined in claim 1 wherein means are provided for mounting said pair of cam followers upon teeth of said driven means tooth segment.
 9. The device as defined in claim 2 wherein said tooth segments are in a third plane spaced from and parallel to said first and second planes.
 10. The device as defined in claim 5 wherein one of said cams and cam followers are in a first plane and another of said cams and cam followers are in a second plane in spaced parallel relationship to said first plane. 